(a) From(31)and(32) ofSection 5.3weknow that whenn = 0, Legendre's differential equation
Chapter 5, Problem 24(choose chapter or problem)
(a) From(31)and(32) ofSection 5.3weknow that whenn = 0, Legendre's differential equation (1 - i2)y" - 2xy' = 0 has the polynomial solution y = P0(x) = 1. Use (5) of Section 3.2 to show that a second Legendre function satisfying the DE on the interval (-1, 1) is 1 (1 + x) y=-ln -- . 2 1 - x (b) We also know from (31) and (32) of Section 5.3 that when n = 1, Legendre's differential equation (1 - i2)y" - 2xy' + 2y = 0 possesses the polynomial solution y = Pi(x) = x. Use (5) of Section 3.2 to show that a second Legendre function satisfying the DE on the interval (-1, 1) is x (1 + x) y = -ln -- - 1. 2 1 - x ( c) Use a graphing utility to graph the logarithmic Legendre functions given in parts (a) and (b).
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